TY - JOUR

T1 - Pseudorandom sets in Grassmann graph have near-perfect expansion

AU - Kho, Subhash

AU - Minzer, Dor

AU - Safra, Muli

N1 - Publisher Copyright:
© 2023 Department of Mathematics, Princeton University.

PY - 2023

Y1 - 2023

N2 - We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. 162 (2005), 439-485], and new hardness gaps for Unique-Games. The Grassmann graph Grglobal contains induced subgraphs Grlocal that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is o(1) inside all subgraphs Grlocal whose order is O(1) lower than that of Grglobal. We prove that pseudorandom sets have expansion 1 - o(1), greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.

AB - We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. 162 (2005), 439-485], and new hardness gaps for Unique-Games. The Grassmann graph Grglobal contains induced subgraphs Grlocal that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is o(1) inside all subgraphs Grlocal whose order is O(1) lower than that of Grglobal. We prove that pseudorandom sets have expansion 1 - o(1), greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.

KW - hypercontractivity

KW - probabilistically checkable proofs

KW - small-set expansion

KW - unique-games conjecture

UR - http://www.scopus.com/inward/record.url?scp=85162969209&partnerID=8YFLogxK

U2 - 10.4007/ANNALS.2023.198.1.1

DO - 10.4007/ANNALS.2023.198.1.1

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AN - SCOPUS:85162969209

SN - 0003-486X

VL - 198

SP - 1

EP - 92

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -